Estimating number of infected people and getting bounds of its probability based on a few samples.

Your expression $ {500choose20} $ is the number of subsets of size $20$ that can be drawn from your set of $500$. The expression $ {mchoose n}{500-mchoose 20-n} $ is, for a given subset of size $ m $, the number of subsets of size $ 20 $ which include exactly $ n $ members of that given subset and exactly $ 20-n $ members of its complement. So, if each subset of size $ 20 $ is equally likely, the fraction $ frac{{mchoose n}{500-mchoose 20-n}}{{500choose20}} $ is the probability that your randomly chosen subset of size $ 20 $ contains exactly $ n $ members of the given subset of size $ m $.

The problem here is that the description of the problem makes $ m $ a non-random number whose value is already determinable, although unknown. Its exact value could be obtained by testing all $500$ blood samples. Non-Bayesians would consider it inappropriate to treat it as a random variable and if they had to estimate its value from a random sample of size $20$ they would probably use some sort of significance test.

That your exam question does treat it as a random variable implies, I presume, that you're required to adopt a Bayesian approach, which would entail your assigning a prior distribution to that random variable. For the moment, let's treat this prior, $ pi $, say, as arbitrary: $$ pi_m=P(M=m) . $$ You can then obtain the posterior distribution of $ M $, given $ N=n $ , from Bayes's theorem: begin{align} P(M=m,| N=n,)&=frac{P(N=n |,M=m,)P(M=m)}{P(N=n)}\ &=cases{ frac{{mchoose n}{500-mchoose 20-n}pi_m}{sum_{i=n}^{480+n} {ichoose n}{500-ichoose 20-n}pi_i}& if $ nle mle n+ 480$\ 0&otherwise .} end{align} Your formula for $ E[M|N=n] $ is correct, except that begin{align} P(M=m, N=n)&=frac{{mchoose n}{500-mchoose 20-n}pi_m}{{500choose20}} text{ and}\ P(N=n)&= displaystylesum_{i=n}^{480+n} frac{{ichoose n}{500-ichoose 20-n}pi_i}{{500choose20}} . end{align} The value of $ Pbig(M>E[M]+1,|,N=4big) $ is given by begin{align} Pbig(M>E[M]+1,|,N=4big)&=sum_{m= max(lfloor E[M] rfloor +2, 4)}^{484}P(M=m,|,N=4,)\ &=frac{sum_{m= max(lfloor sum_{m=0}^{500}mpi_m rfloor +2, 4)}^{484} {mchoose 4}{500-mchoose 16}pi_m}{sum_{i=4}^{484} {ichoose 4}{500-ichoose 16}pi_i} . end{align} Coming to the vexed question of what to choose for $ pi $, the choice of priors in Bayesian statistics is nearly always going to be somewhat subjective. Since I have no idea how this topic was treated in your course, I also have little idea how your examiners would have expected you to handle it in the exam question you've quoted. Also, while I have used statistics professionally, I certainly wouldn't claim to have ever been a professional statistician, let alone one with a good working knowledge of Bayesian statistics. Please bear that in mind while reading the following suggestion.

If your $ 500 $ blood samples were taken somewhat randomly from a large population, it seems to me that a reasonable choice for $ pi $ would be $ text{Binomial}(500,p) $: $$ P(M=m)={500choose m}p^m(1-p)^m $$ for some value of $ p $, which would be the proportion of the large population that are infected. For COVID-$19$, $ p $ will not be known exactly, but if you know the population from which the sample was drawn you may have a reasonable estimate that you could use for the value of $ p $. Otherwise, the best you're likely to be able to do is to get expert epidemiologists to suggest a range $ [a,b] $ in which they think $ p $ is $90%$ (say) likely to lie, and choose a suitable prior distribution $ Pi $ for $ p $ such that $ Pibig([a,b]big)=0.9 $. Your prior distribution for $ M $ will then be $$ P(M=m)={500choose m}int_0^1p^m(1-p)^mdPi(p) , $$ and $ E[M]=500E[p]=500displaystyleint_0^1pdPi(p) $.